# firstQuadrantComplex -- form the first quadrant complex

## Synopsis

• Usage:
firstQuadrantComplex(T,c)
• Inputs:
• T, , a (part of a) Tate resolution on a product of t projective spaces
• c, a list, cohomological degree of the lower corner of the first complex
• Outputs:

## Description

Form the first quadrant complex with corner c of a (part of a) Tate resolution T as defined in Tate Resolutions on Products of Projective Spaces.

 i1 : (S,E) = productOfProjectiveSpaces {1,1}; i2 : T1= (dual res( trim (ideal vars E)^2,LengthLimit=>8))[1]; i3 : T=trivialHomologicalTruncation(T2=res(coker upperCorner(T1,{4,3}),LengthLimit=>13)[7],-5,6); i4 : betti T -5 -4 -3 -2 -1 0 1 2 3 4 5 6 o4 = total: 22 27 26 36 63 98 136 181 236 304 388 491 0: 22 27 18 6 . . . . . . . . 1: . . 8 30 63 98 132 166 200 234 268 302 2: . . . . . . 4 15 36 70 120 189 o4 : BettiTally i5 : cohomologyMatrix(T,-{4,4},{3,2}) o5 = | 27h 20h 13h 6h 1 8 15 22 | | 16h 12h 8h 4h 0 4 8 12 | | 5h 4h 3h 2h h 0 1 2 | | 6h2 4h2 2h2 0 2h 4h 6h 8h | | 17h2 12h2 7h2 2h2 3h 8h 13h 18h | | 28h2 20h2 12h2 4h2 4h 12h 20h 28h | | 39h2 28h2 17h2 6h2 5h 16h 27h 38h | 7 8 o5 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i6 : fqT=firstQuadrantComplex(T,-{2,1}); i7 : betti fqT -5 -4 -3 -2 -1 0 1 o7 = total: 22 27 26 18 22 12 5 0: 22 27 18 6 . . . 1: . . 8 12 22 12 3 2: . . . . . . 2 o7 : BettiTally i8 : cohomologyMatrix(fqT,-{4,4},{3,2}) o8 = | 0 0 13h 6h 1 8 15 22 | | 0 0 8h 4h 0 4 8 12 | | 0 0 3h 2h h 0 1 2 | | 0 0 2h2 0 2h 4h 6h 8h | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | 7 8 o8 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i9 : cohomologyMatrix(fqT,-{2,1},-{1,0}) o9 = | 3h 2h | | 2h2 0 | 2 2 o9 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i10 : lqT=lastQuadrantComplex(T,-{2,1}); i11 : betti lqT 3 4 5 6 o11 = total: 12 37 78 138 2: 12 37 78 138 o11 : BettiTally i12 : cohomologyMatrix(lqT,-{4,4},{3,2}) o12 = | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 | | 17h2 12h2 0 0 0 0 0 0 | | 28h2 20h2 0 0 0 0 0 0 | | 39h2 28h2 0 0 0 0 0 0 | 7 8 o12 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i13 : cohomologyMatrix(lqT,-{3,2},-{2,1}) o13 = | 0 0 | | 12h2 0 | 2 2 o13 : Matrix (ZZ[h, k]) <--- (ZZ[h, k]) i14 : cT=cornerComplex(T,-{2,1}); i15 : betti cT -5 -4 -3 -2 -1 0 1 2 3 4 5 o15 = total: 22 27 26 18 22 12 5 12 37 78 138 0: 22 27 18 6 . . . . . . . 1: . . 8 12 22 12 3 . . . . 2: . . . . . . 2 . . . . 3: . . . . . . . 12 37 78 138 o15 : BettiTally i16 : cohomologyMatrix(cT,-{4,4},{3,2}) o16 = | 0 0 13h 6h 1 8 15 22 | | 0 0 8h 4h 0 4 8 12 | | 0 0 3h 2h h 0 1 2 | | 0 0 2h2 0 2h 4h 6h 8h | | 17h3 12h3 0 0 0 0 0 0 | | 28h3 20h3 0 0 0 0 0 0 | | 39h3 28h3 0 0 0 0 0 0 | 7 8 o16 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])